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3Draft version June 11, 2013Preprint typeset using LATEX style emulateapj v. 5/2/11

THE CO-TO-H2 CONVERSION FACTOR

Alberto D. Bolatto, Mark Wolfire,Department of Astronomy, University of Maryland, College Park, MD 20742

and

Adam K. LeroyNational Radio Astronomy Observatory, Charlottesville, VA 22903

Draft version June 11, 2013

ABSTRACT

CO line emission represents the most accessible and widely used tracer of the molecular interstellarmedium. This renders the translation of observed CO intensity into total H2 gas mass critical tounderstand star formation and the interstellar medium in our Galaxy and beyond. We review thetheoretical underpinning, techniques, and results of efforts to estimate this CO-to-H2 “conversionfactor,” XCO, in different environments. In the Milky Way disk, we recommend a conversion factorXCO = 2×1020 cm−2(K km s−1)−1 with ±30% uncertainty. Studies of other “normal galaxies” returnsimilar values in Milky Way-like disks, but with greater scatter and systematic uncertainty. Departuresfrom this Galactic conversion factor are both observed and expected. Dust-based determinations,theoretical arguments, and scaling relations all suggest that XCO increases with decreasing metallicity,turning up sharply below metallicity ≈ 1/3–1/2 solar in a manner consistent with model predictionsthat identify shielding as a key parameter. Based on spectral line modeling and dust observations,XCO appears to drop in the central, bright regions of some but not all galaxies, often coincident withregions of bright CO emission and high stellar surface density. This lower XCO is also present in theoverwhelmingly molecular interstellar medium of starburst galaxies, where several lines of evidencepoint to a lower CO-to-H2 conversion factor. At high redshift, direct evidence regarding the conversionfactor remains scarce; we review what is known based on dynamical modeling and other arguments.Subject headings: ISM: general — ISM: molecules — galaxies: ISM — radio lines: ISM

1. INTRODUCTION

Molecular hydrogen, H2, is the most abundantmolecule in the universe. With the possible exceptionof the very first generations of stars, star formation is fu-eled by molecular gas. Consequently, H2 plays a centralrole in the evolution of galaxies and stellar systems (seethe recent review by Kennicutt & Evans 2012). Unfor-tunately for astronomers interested in the study of themolecular interstellar medium (ISM), cold H2 is not di-rectly observable in emission. H2 is a diatomic moleculewith identical nucleii and therefore possesses no perma-nent dipole moment and no corresponding dipolar rota-tional transitions. The lowest energy transitions of H2are its purely rotational quadrupole transitions in thefar infrared at λ = 28.22 µm and shorter wavelengths.These are weak owing to their long spontaneous decaylifetimes τdecay ∼ 100 years. More importantly, the twolowest para and ortho transitions have upper level ener-gies E/k ≈ 510 K and 1015 K above ground (Dabrowski1984). They are thus only excited in gas with T & 100 K.The lowest vibrational transition of H2 is even more dif-ficult to excite, with a wavelength λ = 2.22 µm anda corresponding energy E/k = 6471 K. Thus the coldmolecular hydrogen that makes up most of the molecu-lar ISM in galaxies is, for all practical purposes, invisiblein emission.Fortunately, molecular gas is not pure H2. Helium, be-

ing monoatomic, suffers from similar observability prob-lems in cold clouds, but the molecular ISM also containsheavier elements at the level of a few ×10−4 per H nu-cleon. The most abundant of these are oxygen and car-

bon, which combine to form CO under the conditionsprevalent in molecular clouds. CO has a weak perma-nent dipole moment (µ ≈ 0.11 D = 0.11× 10−18 esu cm)and a ground rotational transition with a low excitationenergy hν/k ≈ 5.53 K. With this low energy and crit-ical density (further reduced by radiative trapping dueto its high optical depth), CO is easily excited even incold molecular clouds. At a wavelength of 2.6 mm, theJ = 1 → 0 transition of CO falls in a fairly transparentatmospheric window. It has thus become the workhorsetracer of the bulk distribution of H2 in our Galaxy andbeyond.As a consequence, astronomers frequently employ CO

emission to measure molecular gas masses. The standardmethodology posits a simple relationship between the ob-served CO intensity and the column density of moleculargas, such that

N(H2) = XCO W(12C16O J = 1 → 0), (1)

where the column density, N(H2), is in cm−2 and the

integrated line intensity, W(CO)1, is in traditional radioastronomy observational units of K km s−1. A corollaryof this relation arises from integrating over the emittingarea and correcting by the mass contribution of heavierelements mixed in with the molecular gas,

Mmol = αCO LCO . (2)

1 Henceforth we refer to the most common 12C16O isotopologueas simply CO, and unless otherwise noted to the ground rotationaltransition J = 1 → 0.

http://arxiv.org/abs/1301.3498v3

2

Here Mmol has units of M⊙ and LCO is usually ex-pressed in K km s−1 pc2. LCO relates to the ob-served integrated flux density in galaxies via LCO =2453SCO∆v D

2L/(1 + z), where SCO∆v is the integrated

line flux density, in Jy km s−1, DL is the luminosity dis-tance to the source in Mpc, and z is the redshift (e.g.,Solomon & Vanden Bout 2005, use Eq. 7 to convertbetween W(CO) and SCO∆v). Thus αCO is simply amass-to-light ratio. The correction for the contributionof heavy elements by mass reflects chiefly helium andamounts to a ≈ 36% correction based on cosmologicalabundances.Both XCO and αCO are referred to as the “CO-

to-H2 conversion factor.” For XCO = 2 ×1020 cm−2(K km s−1)−1 the corresponding αCO is4.3 M⊙ (K km s

−1 pc2)−1. To translate integrated fluxdensity directly to molecular mass, Equation 2 can bewritten as

Mmol = 1.05× 104(

XCO

2× 1020 cm−2K km s−1

)

SCO∆v D2L

(1 + z).

(3)For convenience we define

XCO,20 ≡XCO

1× 1020 cm−2(K km s−1)−1 . (4)

We discuss the theoretical underpinnings of these equa-tions in §2.Note that the emission from CO J = 1 → 0 is found

to be consistently optically thick except along very lowcolumn density lines-of-sight, as indicated by ratios of12CO to 13CO intensities much lower than the isotopicratio. The reason for this is simple to illustrate. Theoptical depth of a CO rotational transition is

τJ =8π3

3hµ2

J

gJ

(

ehνJ/kTex − 1) NJ∆ v

, (5)

where J and NJ are the rotational quantum number andthe column density in the upper level of the J → J − 1transition, ν is the frequency, Tex is the excitation tem-perature (in general a function of J , and restricted to bebetween the gas kinetic temperature and that of the Cos-mic Microwave Background), ∆ v is the velocity width, µis the dipole moment, gJ = 2J+1 is the statistical weightof level J , and h and k are the Planck and Boltzmannconstants respectively. Under typical conditions at themolecular boundary, τ ≈ 1 for the J = 1 → 0 transitionrequires N(H2) ≈ 2−3×1020 cm−2 for a Galactic carbongas-phase abundance AC ∼ 1.6×10−4 (Sofia et al. 2004).At the outer edge of a cloud the carbon is mainly C+,which then recombines with electrons to form neutral C(Fig. 1). Carbon is converted to CO by a series of re-actions initiated by the cosmic-ray ionization of H or H2(e.g., van Dishoeck & Black 1988) and becomes the dom-inant carrier of carbon at AV ∼ 1−2. The CO J = 1 → 0line turns optically thick very quickly after CO becomesa significant carbon reservoir, over a region of thickness∆AV ∼ 0.2− 0.3 for a typical Galactic dust-to-gas ratio(c.f., Eq. 21).Equations 1 and 2 represent highly idealized, simplified

relations where all the effects of environment, geometry,

n(X

)/n(H

)

10−8

10−6

10−4

10−2

100

HIH

2

C+

CCO

N(X

)[cm

−2 ]

1014

1016

1018

HI ×10−6

H2 ×10−6

C+

CCO

W(X

)[K

km

s−1 ]

AV

τ C O≈ 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

40

[CII][CI] 1−0CO 1−0

Fig. 1.— Calculated cloud structure as a function of optical depthinto the cloud. Top panel shows the fractional abundance of HI,H2, C+, C, and CO. Middle panel shows their integrated columndensities from the cloud edge. Bottom panel shows the emergentline intensity in units of K km s−1 for [CII] 158 µm, [CI] 609µm, and CO J = 1 → 0. The grey vertical bar shows where COJ = 1 → 0 becomes optically thick. At the outer edge of the cloudgas is mainly HI. H2 forms at AV ∼ 0.5 while the carbon is mainlyC+. The C+ is converted to C at AV ∼ 1 and CO dominates atAV & 2. The model uses constant H density n = 3 × 10

3 cm−3,a radiation field χ = 30 times the interstellar radiation field ofDraine (1978), a primary cosmic-ray ionization rate of 2 × 10−16

s−1 per hydrogen nucleon, and are based on the PDR models ofWolfire, Hollenbach & McKee (2010) and Hollenbach et al. (2012).

excitation, and dynamics are subsumed into the XCOor αCO coefficients. A particular example is the effectthat spatial scales have on the CO-to-H2 conversion fac-tor. Indeed, for the reasons discussed in the previousparagraph, XCO along a line-of-sight through a densemolecular cloud where AV & 10 is not expected to be thesame asXCO along a diffuse line-of-sight sampling mostlymaterial where AV < 1 (see, for example, Pineda et al.2010; Liszt & Pety 2012). Thus on small spatial scaleswe expect to see a large variability in the CO-to-H2 con-version factor. This variability will average out on thelarge spatial scales, to a typical value corresponding tothe dominant environment. Because of the large opticaldepth of the CO J = 1 → 0 transition the velocity disper-sion giving rise to the width of the CO line will also playan important role on W(CO), and indirectly on XCO.Indeed, there is not one value of XCO that is correct andapplicable to each and every situation, although there arevalues with reasonable uncertainties that are applicableover large galactic scales.The plan of this work is as follows: in the remainder of

this section we provide a brief historical introduction. In§2 we present the theoretical background to the CO-to-H2 conversion factor. In §4 we review the methodologyand measurements of XCO in the Milky Way, the bestunderstood environment. We characterize the range ofvalues found and the underlying physics for each mea-surement technique. In §5 we review the literature onXCO determinations in “normal” star-forming galaxiesand discuss the techniques available to estimate XCOin extragalactic systems. In §6 we consider the effect

3

of metallicity, a key local physical parameter. In §7 wereview the measurements and the physical mechanismsaffecting the value of the CO-to-H2 conversion factor inthe starburst environments of luminous and ultralumi-nous galaxies. In §8 we consider the explicit case of XCOin high redshift systems, where a much more restrictedrange of observations exist. In §3 we discuss the resultsof recent calculations of molecular clouds including theeffects of turbulence and chemistry. Finally, in §9 we willoffer some recommendations and caveats as to the bestvalues of XCO to use in different environments, as wellas some suggestions about open avenues of research onthe topic.

1.1. Brief Historical Perspective

Carbon monoxide was one of the first interstel-lar medium molecules observed at millimeter wave-lengths. Wilson, Jefferts & Penzias (1970) reportedthe discovery of intense CO emission from the Orionnebula using the 36 foot NRAO antenna at KittPeak, Arizona. Surveys of molecular clouds in theGalaxy (e.g., Solomon et al. 1972; Wilson et al. 1974;Scoville & Solomon 1975; Burton et al. 1975) establishedmolecular gas to be widespread in the inner Milky Waywith a distribution that resembles giant HII regions moreclosely than that of atomic hydrogen gas. The com-bination of CO and γ-ray observations demonstratedthat H2 dominates over HI by mass in the inner Galaxy(Stecker et al. 1975). By the end of the following decade,these studies extended to complete the mapping of theGalactic plane (Dame et al. 1987).The first extragalactic detections of CO occurred in

parallel with these early Galactic surveys (Rickard et al.1975; Solomon & de Zafra 1975). They found CO to beparticularly bright in galaxies with nuclear activity suchas M 82 and NGC 253. The number of extragalacticCO observations grew rapidly to include several hundredgalaxies over the next two decades (Young & Scoville1991; Young et al. 1995), and CO emission was employedto determine galaxy molecular masses (Young & Scoville1982). By the late 1980s, the first millimeter interferom-eters spatially resolved molecular clouds in other galax-ies (Vogel, Boulanger & Ball 1987; Wilson et al. 1988).Such observations remain challenging, though powerfulnew interferometric facilities such as the Atacama LargeMillimeter Array (ALMA) will change that.The first detection of CO at cosmological red-

shifts targeted ultraluminous infrared sourcesand revealed very large reservoirs of highly ex-cited molecular gas (Brown & Vanden Bout 1991;Solomon, Downes & Radford 1992). Because of the deepintegrations required, the number of high redshift COdetections grew slowly at first (Solomon & Vanden Bout2005), but this field is now developing rapidly drivenby recent improvements in telescope sensitivity (see thereview by Carilli & Walter in this issue). An increasedappreciation of the roles of gas and star formation inthe field of galaxy evolution and the according needto determine accurate gas masses provides one of themotivations for this review.

1.2. CO Excitation

Under average molecular cloud conditions, COmolecules are excited through a combination of colli-

sions with H2 and radiative trapping. They de-excitethrough spontaneous emission and collisions, except atvery high densities where collisions are extremely fre-quent. Neglecting the effect of radiative trapping, ra-diative and collisional de-excitation will balance for acritical density ncr,J ≡ AJ/γJ(Tkin) (neglecting the ef-fects of stimulated emission), where Tkin is the kineticgas temperature. Thus, for n ≫ ncr,J and excitationtemperatures Tex,J ≫ EJ/k ≈ 5.53 J(J + 1)/2 K theupper level of the J → J − 1 transition will be popu-lated and the molecule will emit brightly. In these ex-pressions AJ is the Einstein coefficient for spontaneousemission (only transitions with |∆J | = 1 are allowed),AJ = 64π

4ν3Jµ2J/(3hc3gJ) (A1 ≈ 7.11× 10−8 s−1). The

parameter γJ (T ) is the corresponding collisional coeffi-cient (the sum of all collisional rate coefficients for tran-sitions with upper level J), which is a weak function oftemperature. For CO, γ1 ∼ 3.26×10−11 cm3 s−1 for col-lisions with H2 at Tkin ≈ 30 K (Yang et al. 2010). Tex,Jrefers to the excitation temperature, defined as the tem-perature needed to recover the relative populations of theJ and J − 1 levels from the Boltzmann distribution. Ingeneral, Tex,J will be different for different transitions.The critical density for the CO J = 1 → 0 transition

is n(H2)cr,1 ∼ 2200 cm−3. Higher transitions requirerapidly increasing densities and temperatures to be ex-cited, as ncr,J ∝ J3 andEJ ∝ J2. The high optical depthof the CO emission relaxes these density requirements, asradiative trapping reduces the effective density requiredfor excitation by a factor ∼ 1/τJ (the precise factor cor-responds to an escape probability and is dependent ongeometry).The Rayleigh-Jeans brightness temperature, TJ , mea-

sured by a radio telescope for the J → J − 1 transitionwill be

TJ ≈ 5.53 J(

1− e−τJ)

(

1

e5.53 JTex,J − 1

− 1e

5.53 J2.73(z+1) − 1

)

K .

(6)The final term accounts for the effect of the Cosmic Mi-crowave Background at the redshift, z, of interest. Notethat frequently the Rayleigh-Jeans brightness tempera-ture is referred to as the radiation temperature. Ob-servations with single-dish telescopes usually yield an-tenna temperatures corrected by atmospheric attenu-ation (T ∗A), or main beam temperatures (TMB), suchthat TMB = ηMB T

∗A where ηMB is the main beam effi-

ciency of the telescope at the frequency of the observation(see Kutner & Ulich 1981, for further discussion). TheRayleigh-Jeans brightness temperature TJ is identical toTMB for compact sources, while extended sources maycouple to the antenna with a slightly different efficiency.Extragalactic results, and measurements with interfer-

ometers, are frequently reported as flux densities ratherthan brightness temperatures. The relations betweenflux density (in Jy) and Rayleigh-Jeans brightness tem-perature (in K) in general, and for CO lines, are

SJ ∼= 73.5× 10−3 λ−2 θ2 TJ ≈ 10.9× 10−3 J2 θ2 TJ , (7)where θ is the half-maximum at full width of the tele-scope beam (in arcsec), and λ is the wavelength of a

4

transition (in mm).The Rayleigh-Jeans brightness temperature, TJ , ex-

citation temperature, Tex,J , and kinetic temperature,Tkin, are distinct but related. The

12CO transitions usu-ally have τJ ≫ 1, making the Rayleigh-Jeans bright-ness temperature a probe of the excitation temperature,TJ ∼ Tex,J for Tex,J ≫ 5.53J K. In general Tex,J can beshown to be in the range Tcmb ≤ Tex,J ≤ Tkin. At densi-ties much higher than ncr,J the population of the levels Jand lower will approach a Boltzmann distribution, andbecome “thermalized” at the gas kinetic temperature,Tex,J ≈ Tkin. The corresponding Rayleigh-Jeans bright-ness can be computed using Eq. 6. When Tex,J < Tkin,usually Tex,J/Tex,J−1 < 1 for lines arising in the sameparcel of gas, and the excitation of the J level is “subther-mal” (note that this is not equivalent to TJ/TJ−1 < 1,as is sometimes used in the literature).

2. THEORETICAL BASIS

At its core the XCO factor represents a valiant effort touse the bright but optically thick transition of a molecu-lar gas impurity to measure total molecular gas masses.How and why does XCO work?

2.1. Giant Molecular Clouds

Because the 12CO J = 1 → 0 transition is generally op-tically thick, its brightness temperature is related to thetemperature of the τCO = 1 surface, not the column den-sity of the gas. Information about the mass of a self-gravitating entity, such as a molecular cloud, is conveyedby its line width, which reflects the velocity dispersion ofthe emitting gas.A simple and exact argument can be laid for virialized

molecular clouds, that is clouds where twice the inter-nal kinetic energy equals the potential energy. FollowingSolomon et al. (1987), the virial mass Mvir of a giantmolecular cloud (GMC) in M⊙ is

Mvir =3(5− 2k)G(3 − k) Rσ

2, (8)

where R is the projected radius (in pc), σ is the 1D

velocity dispersion (in km s−1; σ3D =√3σ), G is the

gravitational constant (G ≈ 1/232 M−1⊙ pc km2 s−2),and k is the power-law index of the spherical volumedensity distribution, ρ(r) ∝ r−k . The coefficient infront of Rσ2 is only weakly dependent on the densityprofile of the virialized cloud, and corresponds to ap-proximately 1160, 1040, and 700 for k = 0, 1, and 2respectively (MacLaren, Richardson & Wolfendale 1988;Bertoldi & McKee 1992). Unless otherwise specified, weadopt k = 1 for the remainder of the discussion. Thisexpression of the virial mass is fairly robust if otherterms in the virial theorem (McKee & Zweibel 1992;Ballesteros-Paredes 2006), such as magnetic support, canbe neglected (for a more general expression applicableto spheroidal clouds and a general density distributionsee Bertoldi & McKee 1992). As long as molecular gasis dominating the mass enclosed in the cloud radius andthe cloud is approximately virialized, Mvir will be a goodmeasure of the H2 mass.Empirically, molecular clouds are observed to follow a

size-line width relation (Larson 1981; Heyer et al. 2009)such that approximately

σ = C R0.5, (9)

with C ≈ 0.7 km s−1 pc−0.5 (Solomon et al. 1987;Scoville et al. 1987; Roman-Duval et al. 2010). This re-lation is an expression of the equilibrium supersonic tur-bulence conditions in a highly compressible medium,and it is thought to apply under very general condi-tions (see §2.1 of McKee & Ostriker 2007, for a discus-sion). In fact, to within our current ability to measurethese two quantities such a relation is also approximatelyfollowed by extragalactic GMCs in galaxy disks (e.g.,Rubio, Lequeux & Boulanger 1993; Bolatto et al. 2008;Hughes et al. 2010).Note that insofar as the size dependence of Eq. 9 is

close to a square root, the combination of Eqs. 8 and 9yields that Mvir ∝ σ4, and molecular clouds that fulfillboth relations have a characteristic mean surface density,ΣGMC, at a value related to the coefficient of Eq. 9 sothat ΣGMC = Mvir/πR

2 ≈ 331C2 for our chosen densityprofile ρ ∝ r−1. We return to the question of ΣGMC inthe Milky Way in §4.4.Since the CO luminosity of a cloud, LCO, is the prod-

uct of its area (π R2) and its integrated surface brightness

(TB√2π σ), then LCO =

√2π3TB σ R

2, where TB is theRayleigh-Jeans brightness temperature of the emission(see §1.2). Using the size-line width relation (Equation9) to substitute for R implies that LCO ∝ TBσ5. Em-ploying this relation to replace σ in Mvir ∝ σ4 we obtaina relation between Mvir and LCO,

Mvir ≈ Mmol ≈ 200(

C1.5 LCOTB

)0.8

. (10)

That is, for GMCs near virial equilibrium with approxi-mately constant brightness temperature, TB, we expectan almost linear relation between virial mass and lumi-nosity. The numerical coefficient in Equation 10 is onlya weak function of the density profile of the cloud. Thenusing the relation between C and ΣGMC we obtain thefollowing expression for the conversion factor

αCO ≡MmolLCO

≈ 6.1LCO−0.2 T−0.8B ΣGMC0.6. (11)

Equations 10 and 11 rest on a number of assumptions.We assume 1) virialized clouds with, 2) masses domi-nated by H2 that 3) follow the size-line width relationand 4) have approximately constant temperature. Equa-tion 11 applies to a single, spatially resolved cloud, asΣGMC is the resolved surface density.We defer the discussion of the applicability of the virial

theorem to §4.1.1, and the effect of other mass compo-nents to §2.3. The assumption of a size-line width rela-tion relies on our understanding of the properties of tur-bulence in the interstellar medium. The result σ ∝

√R

follows our expectations for a highly compressible tur-bulent flow, with a turbulence injection scale at leastcomparable to GMC sizes. The existence of a narrowrange of proportionality coefficients, corresponding to asmall interval of GMC average surface densities, is lesswell understood (for an alternative view on this point,see Ballesteros-Paredes et al. 2011). In fact, this narrow

5

range could be an artifact of the small dynamic range ofthe samples (Heyer et al. 2009). Based on observationsof the Galactic center (Oka et al. 2001) and starburstgalaxies (e.g., Rosolowsky & Blitz 2005), ΣGMC likelydoes vary with environment. Equation 11 implies thatany such systematic changes in ΣGMC will also lead tosystematic changes in XCO, though in actual starburstenvironments the picture is more complex than impliedby Equation 11. Section 7 reviews the case of bright,dense starbursts in detail.This calculation also implies a dependence of XCO on

the physical conditions in the GMC, density and tem-perature. Combining Eqs. 8 and 9 with Mvir ∝ ρR3,where ρ is the gas density, yields σ ∝ ρ−0.5. Meanwhile,because CO emission is optically thick the observed lu-minosity depends on the brightness temperature, TB, aswell as the line width, so that LCO ∝ σTB. Substitutingin the relationship between density and line width,

αCO ∝ρ0.5

TB. (12)

The brightness temperature, TB, will depend on the ex-citation of the gas (Eq. 6) and the filling fraction ofemission in the telescope beam, fb. For high density andoptical depth the excitation temperature will approachthe kinetic temperature. Under those conditions, Eq. 12also implies αCO ∝ ρ0.5(fb Tkin)−1.Thus, even for virialized GMCs we expect that the

CO-to-H2 conversion factor will depend on environmen-tal parameters such as gas density and temperature. Tosome degree, these dependencies may offset each other.If denser clouds have higher star formation activity andare consequently warmer, the opposite effects of ρ andTB in Eq. 12 may partially cancel yielding a conversionfactor that is closer to a constant than we might other-wise expect.We also note that relation between mass and luminos-

ity expressed by Eq. 10 is not exactly linear, which isthe reason for the weak dependence of αCO on LCO orMmol in Eq. 11. As a consequence, even for GMCs thatobey this simple picture, αCO will depend (weakly) onthe mass of the cloud considered, varying by a factor of∼ 4 over 3 orders of magnitude in cloud mass.

2.2. Galaxies

This simple picture for how the CO luminosity can beused to estimate masses of individual virialized cloudsis not immediately applicable to entire galaxies. An ar-gument along similar lines, however, can be laid out tosuggest that under certain conditions there should bean approximate proportionality between the integratedCO luminosity of entire galaxies and their molecularmass. This is known as the “mist” model, for rea-sons that will become clear in a few paragraphs. Fol-lowing Dickman, Snell & Schloerb (1986), the luminos-ity due to an ensemble of non-overlapping CO emit-ting clouds is LCO ∝

∑

i ai TB(ai)σi, where ai is thearea subtended by cloud i, and TB(ai) and σi are itsbrightness temperature and velocity dispersion, respec-tively. Under the assumption that the brightness tem-perature is mostly independent of cloud size, and thatthere is a well-defined mean, TB, then TB(ai) ≈ TB.We can rewrite the luminosity of the cloud ensemble

as LCO ≈√2πTB Nclouds < πR

2i σ(Ri) >, where the

brackets indicate expectation value, Nclouds is the num-ber of clouds within the beam, and we have used ai =π R2i . Similarly, the total mass of gas inside the beam isMmol ≈ Nclouds < 4/3πR3i ρ(Ri) >, where ρ(Ri) is thevolume density of a cloud of radius Ri. Using our defi-nition from Eq. 2 and dropping the i indices, it is thenclear that

αCO ≡MmolLCO

≈√

8

9π

< R3ρ(R) >

TB < R2σ(R) >. (13)

If the individual clouds are virialized they will followEq. 8, or equivalently σ = 0.0635R

√ρ. Substituting

into Eq. 13 we find

αCO ∝< R3ρ >

TB < R3ρ0.5 >, (14)

which is analogous to Eq. 12 (obtained for individualclouds). As Dickman, Snell & Schloerb (1986) discuss,it is possible to generalize this result if the clouds in agalaxy follow a size-line width relation and they have aknown distribution of sizes. Assuming individually viri-alized clouds, and using the size-line width relation (Eq.9), we can write down Eq. 13 as

αCO = 7.26

√ΣGMCTB

< R2 >

< R2.5 >, (15)

where we have introduced the explicit dependence of thecoefficient of the size-line width relation on the cloudsurface density, ΣGMC. This equation is the analogue ofEq. 11.In the context of these calculations, CO works as a

molecular mass tracer in galaxies because its intensity isproportional to the number of clouds in the beam, andbecause through virial equilibrium the contribution fromeach cloud to the total luminosity is approximately pro-portional to its mass, as discussed for individual GMCs.This is the essence of the “mist” model: although eachparticle (cloud) is optically thick, the ensemble acts opti-cally thin as long as the number density of particles is lowenough to avoid shadowing each other in spatial-spectralspace.Besides the critical assumption of non-overlapping

clouds, which could be violated in environments of veryhigh density leading to “optical depth” problems thatmay render CO underluminous, the other key assump-tion in this model is the virialization of individual clouds,already discussed for GMCs in the Milky Way. The ap-plicability of a uniform value of αCO across galaxies relieson three assumptions that should be evident in Eq. 15: asimilar value for ΣGMC, similar brightness temperaturesfor the CO emitting gas, and a similar distribution ofGMC sizes that determines the ratio of the expectationvalues < R2 > / < R2.5 >. Very little is known cur-rently on the distribution of GMC sizes outside the Lo-cal Group (Blitz et al. 2007; Fukui & Kawamura 2010),and although this is a potential source of uncertainty inpractical terms this ratio is unlikely to be the dominantsource of galaxy-to-galaxy variation in αCO.

2.3. Other Sources of Velocity Dispersion

6

Because CO is optically thick, a crucial determinantof its luminosity is the velocity dispersion of the gas,σ. In our discussion for individual GMCs and ensem-bles of GMCs in galaxies we have assumed that σ isultimately determined by the sizes (through the size-line width relation) and virial masses of the clouds. Itis especially interesting to explore what happens whenthe velocity dispersion of the CO emission is relatedto an underlying mass distribution that includes othercomponents besides molecular gas. Following the rea-soning by Downes, Solomon & Radford (1993) (see alsoMaloney & Black 1988; Downes & Solomon 1998) andthe discussion in §2.1, we can write a cloud luminos-ity LCO for a fixed TB as L

∗CO = LCO σ

∗/σ, where theasterisk indicates quantities where the velocity disper-sion of the gas is increased by other mass components,such as stars. Assuming that both the molecular gasand the total velocity dispersion follow the virial ve-locity dispersion due to a uniform distribution of mass,σ =

√

GM/5R, then L∗CO = LCO√

M∗/Mmol, whereM∗ represents the total mass within radius R. Substi-tuting LCO = Mmol/αCO yields the result αCOL

∗CO =√

Mmol M∗.Therefore, the straightforward application of αCO to

the observed luminosity L∗CO will yield an overestimateof the molecular gas mass, which in this simple reason-ing will be the harmonic mean of the real molecular massand the total enclosed mass. If the observed velocity dis-persion is more closely related to the circular velocity, asmay be in the center of a galaxy, then σ∗ ≈

√

GM∗/Rand the result of applying αCO to L

∗CO will be an even

larger overestimate of Mmol. The appropriate value ofthe CO-to-H2 conversion factor to apply under these cir-cumstances in order to correctly estimate the molecularmass is

α∗CO =MmolLCO

∗ = αCO

√

MmolKM∗ (16)

where K is a geometrical correction factor accountingfor the differences in the distributions of the gas andthe total mass, so that K ≡ (σ∗/σ)2. In the extremecase of a uniform distribution of gas responding to thepotential of a rotating disk of stars in a galaxy center,K ∼ 5. Everything else being equal, in a case whereM∗ ∼ 10Mmol the straight application of a standardαCO in a galaxy center may lead to overestimating Mmolby a factor of ∼ 7.Note that for this correction to apply the emission

has to be optically thick throughout the medium. Oth-erwise any increase in line width is compensated by adecrease in brightness, keeping the luminosity constant.Thus this effect is only likely to manifest itself in regionsthat are already rich in molecular gas. Furthermore, itis possible to show that an ensemble of virialized cloudsthat experience cloud-cloud shadowing cannot explain alower XCO, simply because there is a maximum attain-able luminosity. Therefore we expect XCO to drop inregions where the CO emission is extended throughoutthe medium, and not confined to collections of individualself-gravitating molecular clouds. This situation is likelypresent in ultra-luminous infrared galaxies (ULIRGs),where average gas volume densities are higher than the

typical density of a GMC in the Milky Way, suggestinga pervading molecular ISM (e.g., Scoville, Yun & Bryant1997). Indeed, the reduction of XCO in mergers andgalaxy centers has been modeled in detail by Shetty et al.(2011b) and Narayanan et al. (2011, 2012), and directlyobserved (§5.2,§5.3, §7.1, §7.2)

2.4. Optically Thin Limit

Although commonly the emission from 12COJ = 1 → 0 transition is optically thick, under con-ditions such as highly turbulent gas motions orotherwise large velocity dispersions (for example stellaroutflows and perhaps also galaxy winds) then emissionmay turn optically thin. Thus it is valuable to considerthe optically thin limit on the value of the CO-to-H2conversion factor. Using Eq. 5, the definition of opticallythin emission (IJ = τJ [BJ(Tex)−BJ(Tcmb)], whereBJ is the Planck function at the frequency νJ of theJ → J − 1 transition, Tex is the excitation temperature,and Tcmb is the temperature of the Cosmic MicrowaveBackground), and the definition of antenna temperatureTJ , IJ = (2kν

2J/c

2)TJ , the integrated intensity of theJ → J − 1 transition can be written as

W(CO) = TJ∆v =8π3νJ3k

µ2J

gJfcmbNJ . (17)

The factor fcmb accounts for the effect of the Cos-mic Microwave Background on the measured intensity,fcmb = 1 − (ehνJ/kTex − 1)/(ehνJ/kTcmb − 1). Note thatfcmb ∼ 1 for Tex ≫ Tcmb.The column density of H2 associated with this in-

tegrated intensity is simply N(H2) =1

ZCO

∑∞

J=0 NJ ,where ZCO is the CO abundance relative to molecu-lar hydrogen, ZCO = CO/H2. For a Milky Way gasphase carbon abundance, and assuming all gas-phasecarbon is locked in CO molecules, ZCO ≈ 3.2 × 10−4(Sofia et al. 2004). Note, however, that what matters isthe integrated ZCO along a line-of-sight, and CO maybecome optically thick well before this abundance isreached (for example, Fig. 1). Indeed, Sheffer et al.(2008) analyze ZCO in Milky Way lines-of-sight, find-ing a steep ZCO ≈ 4.7× 10−6(N(H2)/1021 cm−2)2.07 forN(H2) > 2.5 × 1020 cm−2, with an order of magnitudescatter (see also Sonnentrucker et al. 2007).When observations in only a couple of transitions

are available, it is useful to assume local thermody-namic equilibrium applies (LTE) and the system is de-scribed by a Boltzmann distribution with a single tem-perature. In that case the column density will beN(CO) = Q(Tex)e

E1/kTexN1/g1, where E1 is the en-ergy of the J = 1 state (E1/k ≈ 5.53 K for CO), andQ(Tex) =

∑∞

J=0 gJe−EJ/kTex corresponds to the parti-

tion function at temperature Tex which can be approx-imated as Q(Tex) ∼ 2kTex/E1 for rotational transitionswhen Tex ≫ 5.5 K (Penzias 1975, note this is accurateto ∼ 10% even down to Tex ∼ 8 K). Using Eq. 17 we canthen write

XCO =N(H2)

W(CO)≈ 1

ZCO

6h

8π3µ2fcmbE1/k

(

TexE1/k

)

eE1/kTex .

(18)

7

Consequently, adopting ZCO = 10−4 and using a repre-

sentative Tex = 30 K, we obtain

XCO ≈ 1.6× 1019Tex30K

e5.53KTex

−0.184 cm−2(K km s−1)−1,

(19)or αCO ≈ 0.34 M⊙ (K km s−1 pc2)−1. These are an or-der of magnitude smaller than the typical values of XCOand αCO in the Milky Way disk, as we will discuss in§4. Note that they are approximately linearly dependenton the assumed ZCO and Tex (for Tex ≫ 5.53 K). Fora similar calculation that also includes an expression fornon-LTE, see Papadopoulos et al. (2012).

2.5. Insights from Cloud Models

A key ingredient in further understanding XCO inmolecular clouds is the structure of molecular cloudsthemselves, which plays an important role in the radia-tive transfer. This is important both for the photodis-sociating and heating ultraviolet radiation, and for theemergent intensity of the optically thick CO lines.The CO J = 1 → 0 transition arises well within the

photodissociation region (PDR) in clouds associated withmassive star formation, or even illuminated by the gen-eral diffuse interstellar radiation field (Maloney & Black1988; Wolfire, Hollenbach & Tielens 1993). At thosedepths gas heating is dominated by the grain photoelec-tric effect whereby stellar far-ultraviolet photons are ab-sorbed by dust grains and eject a hot electron into thegas. The main parameter governing grain photoelectricheating is the ratio χT 0.5kin/ne, where χ is a measure ofthe far-ultraviolet field strength, and ne is the electrondensity. This process will produce hotter gas and higherexcitation in starburst galaxies. At the high densities ofextreme starbursts, the gas temperature and CO exci-tation may also be enhanced by collisional coupling be-tween gas and warm dust grains.Early efforts to model the CO excitation and luminos-

ity in molecular clouds using a large velocity gradientmodel were carried out by Goldreich & Kwan (1974).The CO luminosity-gas mass relation was investigatedby Kutner & Leung (1985) using microturbulent mod-els, and by Wolfire, Hollenbach & Tielens (1993) usingboth microturbulent and macroturbulent models. In mi-croturbulent models the gas has a (supersonic) isotropicturbulent velocity field with scales smaller than the pho-ton mean free path. In the macroturbulent case the scalesize of the turbulence is much larger than the photonmean free path, and the emission arises from separateDoppler shifted emitting elements. Microturbulent mod-els produce a wide range of CO J = 1 → 0 profile shapes,including centrally peaked, flat topped, and severely cen-trally self-reversed, while most observed line profiles arecentrally peaked. Macroturbulent models, on the otherhand, only produce centrally peaked profiles if there area sufficient number of “clumps” within the beam withdensities n & 103 cm−3 in order to provide the peakbrightness temperature. Falgarone et al. (1994) demon-strated that a turbulent velocity field can produce bothpeaked and smooth line profiles, much closer to observa-tions than macroturbulent models.Wolfire, Hollenbach & Tielens (1993) use PDR mod-

els in which the chemistry and thermal balance was cal-

culated self-consistently as a function of depth into thecloud. The microturbulent PDR models are very suc-cessful in matching and predicting the intensity of low-JCO lines and the emission from many other atomic andmolecular species (Hollenbach & Tielens 1997, 1999).For example, the nearly constant ratio of [CII]/COJ = 1 → 0 observed in both Galactic and extragalacticsources (Crawford et al. 1985; Stacey et al. 1991) wasfirst explained by PDR models as arising from high den-

sity (n >∼ 103 cm−3) and high UV field (χ >∼ 103) sourcesin which both the [CII] and CO are emitted from thesame PDR regions in molecular cloud surfaces. Thesemodels show that the dependence on CO luminosity withincident radiation field is weak. This is because as thefield increases the τCO = 1 surface is driven deeper intothe cloud where the dominant heating process, grain pho-toelectric heating, is weaker. Thus the dissociation of COin higher fields regulates the temperature where τCO = 1.PDR models have the advantage that they calculate

the thermal and chemical structure in great detail, sothat the gas temperature is determined where the COline becomes optically thick. We note however, that al-though model gas temperatures for the CO J = 1 → 0line are consistent with observations the model tempera-tures are typically too cool to match the observed high-JCO line emission (e.g., Habart et al. 2010). The modeldensity structure is generally simple (constant density orconstant pressure), and the velocity is generally consid-ered to be a constant based on a single microturbulentvelocity. More recent dynamical models have started tocombine chemical and thermal calculations with full hy-drodynamic simulations.

3. A MODERN THEORETICAL PERSPECTIVE

In recent years there has been much progress in updat-ing PDR models, and in combining hydrodynamic simu-lations, chemical modeling, and radiation transfer codes.

3.1. Photodissociation Regions

Since CO production mainly occurs through ion-neutral chemistry (van Dishoeck & Black 1988) the ion-ization structure through the PDR is important in set-ting the depth of the CO formation. Recombination ofmetal ions on PAHs can modify the abundance of freeelectrons. For a typical GMC with n ∼ 103 cm−3 il-luminated by a radiation field ∼ 10 times the GalacticISRF (the interstellar radiation field in the vicinity of theSun), the τCO = 1 surface is at a depth of AV ∼ 1 whenPAHs are included. Without PAHs the electron abun-dance stays high, and the ion-neutral chemistry is sloweddue to H+3 recombination. Consequently, the τCO = 1surface is pushed deeper into the cloud (AV ∼ 2). ThePAH rates are estimated by Wolfire et al. (2008) by con-sidering the C0/C+ ratio in diffuse lines of sight, butthere is considerable uncertainty both in the rates (Liszt2011) and in the PAH abundance and their variation withcloud depth.Bell et al. (2006) carry out a parameter study of XCO

using PDR models with constant microturbulent linewidth. The calculated XCO versus AV plots have a char-acteristic shape with high XCO at low AV , dropping toa minimum value at AV ∼ 2 − 4 and then slowly ris-ing for increasing AV . The XCO dependence at low AV

8

arises from molecular gas with low CO abundance. Af-ter the CO line intensity becomes optically thick, XCOslowly rises again as N(H2) increases. Increasing densityup to the critical density of CO J = 1 → 0 (n(H2)cr,1 ∼2200 cm−3) enhances the CO excitation and causes XCOto drop. In addition, the minimum moves closer to thecloud surface as does the τCO = 1 surface. An increasein XCO by a factor of more than 100 is found when de-creasing dust and metallicity to 1% of the local GalacticISM, reflecting the larger column of H2 at a given AV .Bell, Viti & Williams (2007) suggest that the appropri-ate XCO value to use in various extragalactic environ-ments can be estimated from the minimum in the XCOversus AV plot.Wolfire, Hollenbach & McKee (2010) use an updated

version of their PDR models, including a self-consistentcalculation for the median density expected from a tur-bulent density distribution. The models provide a the-oretical basis for predicting the molecular mass fractionoutside of the CO emitting region in terms of incidentradiation field and gas metallicity. They assume thatthe median density is given by 〈n〉med = n̄ exp(µ), wheren̄ is the volume averaged density distribution n̄ ∝ 1/r,and µ = 0.5 ln(1+ 0.25M2) (Padoan, Jones & Nordlund1997). The sound speed that enters in the Mach number,M, is calculated from the PDR model output while theturbulent velocity is given by the size-linewidth relation(Eq. 9).

3.2. Numerical Simulations

As an alternative to PDR models with simple geome-tries and densities, hydrodynamical models can be usedto calculate the line width, and density, but with onlylimited spatial resolution and approximate chemistry andthermal balance. Glover & Mac Low (2007,b) carry outsimulations of the formation of molecular clouds usinga modified version of the magnetohydrodynamical codeZEUS-MP. They include a time-dependent chemistry, inparticular for H2 formation and destruction, and ther-mal balance. Glover et al. (2010) enhance the code toaccount for CO chemistry in a turbulent GMC. A turbu-lent picture of a GMC is not one with well defined clumpssurrounded by an interclump medium but one with acontinuous density distribution and constant mixing be-tween low and high density regions. The CO abundancevaries within the cloud depending on the gas density andpenetration of the dissociating radiation field. Calcu-lations of XCO are carried out by Glover & Mac Low(2011), and Shetty et al. (2011,b). The latter two ref-erences include non-LTE CO excitation and line trans-fer in the LVG approximation. These authors carry out3D turbulent simulations in a box of fixed size (20 pc),and turbulence generated with uniform power betweenwavenumbers 1 ≤ k ≤ 2, but with various initial den-sities and metallicities. For their standard cloud modelthe saturation amplitude of the 1D velocity dispersionis 2.4 km s−1. Each line of sight has a different CO in-tensity, and CO and H2 column density depending onthe past and present physical conditions along it. Thus,for a given N(H2) there is a range in XCO values (seeFigs. 5 and 6 in Shetty et al. 2011). The dispersion gen-erally increases for clouds with lower density and lowermetallicity.

Figure 7 in Shetty et al. (2011) shows the calculatedmean XCO in different AV bins for several initial densi-ties and metallicities. The variation in XCO with AV isqualitatively similar to that shown in Bell et al. (2006)for microturbulent models. Lower densities and metallic-ities drive XCO to higher values due to lower CO excita-tion and CO/H2 ratios respectively. At AV & 7 the mod-els with different initial densities converge as the CO linebecomes optically thick. The minimum in XCO occursat larger AV in the hydrodynamic simulations comparedto the one sided PDR models, likely due to the dissoci-ating radiation incident on all sides of the box and dueto its greater penetration along low density lines of sight.A range in density n = 100− 1000 cm−3 and metallicityZ = 0.1−1 Z⊙ produces a range of only XCO,20 ∼ 2−10.Thus although there can be large variations inXCO alongdifferent lines of sight in the cloud the emission weightedXCO is close to the typical Galactic value (§4). In ad-dition, the low metallicity case has higher XCO at lowAV but approaches the Galactic value for higher density(higher AV ) lines of sight. Although the models were runwith constant box size, the dependence on AV suggeststhat clouds with sufficiently small size so that CO doesnot become optically thick will have higherXCO. The re-sults seem to confirm the suggestion by Bell et al. (2006)that the mean XCO value should be near the minimumwhen plotted as XCO versus AV .Shetty et al. (2011b) investigate the results of vary-

ing the temperature, CO abundance, and turbulent linewidth. They find only a weak dependence on temper-ature with XCO ∝ T−0.5kin for 20 K< Tkin

9

of ∼ 2.Global models ofXCO using hydrodynamic simulations

to investigate the variation due to galactic environmentare carried out by Feldmann, Gnedin & Kravtsov (2012)and Narayanan et al. (2011, 2012). A significant problemin numerical simulations is how to handle the physicalconditions and/or line emission within regions smallerthan the spatial grid. Feldmann, Gnedin & Kravtsov(2012) use high resolution (∼ 0.1 pc) simulations fromGlover & Mac Low (2011) for “sub-grid” solutions tocosmological simulations of 60 pc resolution. These mod-els used constant Tkin = 10 K, LTE excitation for CO,with either constant CO line width or one proportionalto Σmol

0.5. By comparing results at their highest res-olutions with those at 1 kpc, and 4 kpc, they assesthe effects of spatial averaging on XCO. The averag-ing tends to reduce the variation of XCO on N(H2) andUV radiation field intensity. At greater than kilopar-sec scales, and H2 column densities between 10

21 cm−2

and 1023 cm−2, XCO changes by only a factor of 2.They find essentially no variation in XCO with UV fieldstrength between 0.1 and 100 times the Galactic ISRF.Feldmann, Gnedin & Kravtsov (2012) do find a signif-icant variation with metallicity, with a scaling XCO∝Z−0.5 for virial line widths. Note that this relation ismuch shallower than found by, for example, Genzel et al.(2012, §8.2). The authors suggest that low metallicityhigh-redshift galaxies may not obey the same gas surface-density to star formation relation observed in local disks.The results of Narayanan et al. (2011) and

Narayanan et al. (2012) will be discussed in §7.2.Here we note that their effective resolution of ∼ 70 pcrequires adopting sub-grid cloud properties. Althoughthe surfaces of GMCs more massive than ∼ 7 × 105M⊙ are resolved, their internal structure is not. Theresulting line and continuum transfer, and temperatureand chemical structure can only be approximate. Theresolution problem also enters in simulations of individ-ual clouds. We showed in §1 that CO becomes opticallythick within a column N(H2) ≈ 2 − 3 × 1020 cm−2.Thus for sub parsec resolution (∼ 0.1 pc) and densitygreater than n & 103 cm−3, the physical conditions(temperature, density, and abundances) are averagedover the line forming region and the calculated emittedintensity can be in error. The resolution problem ismuch more severe for H2 dissociation. Self-shielding ofH2 starts within a column of only N(H2) ∼ 1014 cm−2(de Jong, Boland & Dalgarno 1980). Thus with aresolution of ∼ 0.1 pc, and density of n & 10 cm−3, theoptical depth to dissociating radiation is already τ & 104

in a resolution element. A complementary approach isto apply a state-of-the-art PDR code with well resolvedH2 formation to the output of hydrodynamic simulations(Levrier et al. 2012). With increasing computing powerthe issues of resolution will continue to improve. Wesuggest that high priority should be placed on creatinglarge scale galactic simulations that are well matchedto small scale simulations with resolved cloud structure.The former provide environmental conditions and cloudboundary conditions while the latter provide the chem-istry and line emission in a realistic turbulent cloud.Expanding the library of GMC models with a range ofcolumn densities, line widths, and external heating, and

thoroughly checking them against observations, wouldbe most helpful.

4. XCO IN THE MILKY WAY

The Galaxy is the only source where it is possible todetermine the CO-to-H2 conversion factor in a variety ofways. It thus provides the prime laboratory to investi-gate the calibration and the variations of the proportion-ality between CO emission and molecular mass.In the following sections we will discuss three types of

XCO determinations: 1) employing virial masses, a tech-nique that requires the ability to spatially resolve molec-ular clouds to measure their sizes and kinematics, 2) tak-ing advantage of optically thin tracers of column density,such as dust or certain molecular and atomic lines, and3) using the diffuse γ-ray emission arising from the pionproduction process that takes place when cosmic raysinteract with interstellar medium protons. Gamma-raytechniques are severely limited by sensitivity, and areonly applicable to the Milky Way and the MagellanicClouds. The good level of agreement between these ap-proaches in our own galaxy is the foundation of the useof the CO-to-H2 conversion factor in other galaxies.

4.1. XCO Based on Virial Techniques

The application of the virial theorem to molecularclouds has been discussed by a number of authors, andrecently reviewed by McKee & Ostriker (2007). Herewe just briefly summarize the fundamental points. Thevirial theorem can be expressed in the Lagrangian (fixedmass) or Eulerian (fixed volume) forms, the latter par-ticularly applicable to turbulent clouds where mass isconstantly exchanged with the surrounding medium. Inthe somewhat simpler Lagrangian form, the virial equi-librium equation is

2(K −Ks) +B +W = 0 (20)where K is the volume integral of the thermal plus ki-netic energy, Ks is the surface pressure term, B is thenet magnetic energy including volume and surface terms(which cancel for a completely uniform magnetic field),and W is the net gravitational energy which is deter-mined by the self-generated gravitational potential if theacceleration due to mass external to the cloud can be ne-glected. In the simple case of a uniform, unmagnetizedsphere virial equilibrium implies 2K+W = 0. It is usefulto define the virial parameter, avir , which corresponds tothe ratio of total kinetic energy to gravitational energy(Bertoldi & McKee 1992), so that avir ≡ 5Rσ2/GM .

4.1.1. Are Clouds Virialized?

In this context, gravitationally bound objects haveavir ≃ 1. Whether interstellar clouds are entities in virialequilibrium, even in a time or ensemble averaged sense(McKee 1999), is a matter of current debate. Observa-tional evidence can be interpreted in terms of systemsout of equilibrium with rapid star formation and subse-quent disruption in a few Myr (e.g., Elmegreen 2000), anevolutionary progression and a typical lifetime of a fewtens of Myr, long enough for clouds to become virialized(e.g., Blitz & Shu 1980b; Fukui & Kawamura 2010), or alifetime of hundreds of Myr (e.g., Scoville & Hersh 1979).Roman-Duval et al. (2010) find a median avir ≈ 0.5 for

10

clouds in the inner Galaxy, suggesting that they arebound entities where Mvir represents a reasonable mea-sure of the molecular mass, although casting doubt onthe assumption of exact virial equilibrium. Wong et al.(2011) estimate a very large scatter in avir in theLarge Magellanic Cloud, but do lack an independentmass tracer so their results rest on the assumption ofa fixed XCO. Observations in the outer Galaxy show an-other angle of the situation. Heyer, Carpenter & Snell(2001) find that clouds with Mmol > 10

4 M⊙ are self-gravitating, while small clouds with masses Mmol < 10

3

M⊙ are overpressured with respect to their self-gravity,that is, have avir ≫ 1 and are out of equilibrium. Giventhe observed mass function, however, such clouds repre-sent a very small fraction of the molecular mass of theMilky Way.In any case, observed GMC properties can be under-

stood as a consequence of approximate energy equipar-tition, which observationally is very difficult to distin-guish from virial equilibrium (Ballesteros-Paredes 2006).Clouds with an excess of kinetic energy, avir ≫ 1, per-haps due to ongoing star formation or SNe would berapidly dissipated, while clouds with a dearth of kineticenergy, avir ≪ 1, would collapse at the free-fall velocitywhich is within 40% of the equipartition velocity disper-sion and challenging to distinguish from turbulent mo-tions in observations. Furthermore, the resulting starformation will inject energy into the cloud acting to re-store the balance. Thus from the standpoint of determin-ing cloud masses over large samples, the assumption ofvirial equilibrium even if not strictly correct, is unlikelyto be very wrong.

4.1.2. Observational Results

The most significant study of the relation betweenvirial mass and LCO (the mass-luminosity relation) inthe Milky Way is that by Solomon et al. (1987), whichencompasses 273 clouds and spans several orders of mag-nitude in cloud luminosity and mass. It is dominated byclouds located in the inner Galaxy, in the region of theso-called Molecular Ring, a feature in the molecular sur-face density of the Milky Way peaking at RGC ≈ 4 kpcgalactocentric radius. It uses kinematic distances withan old value of the distance to the Galactic Center,R⊙ = 10 kpc. We report new fits after a 0.85 scal-ing in all distances and sizes and 0.72 in luminositiesto bring them into agreement with the modern distancescale (R⊙ = 8.5 kpc). The virial mass computationsassume a ρ(r) ∝ r−1 (see §2.1).Solomon et al. (1987) find a very strong correlation be-

tweenMvir and LCO, such thatMvir = 37.9LCO0.82 with

a typical dispersion of 0.11 dex for Mvir. Note the excel-lent agreement with the expected mass-luminosity rela-tion in Eq. 10 using a typical CO brightness temperatureTB ≈ 4 K (Maloney 1990). For a cloud at their approxi-mate median luminosity, LCO ≈ 105 K km s−1 pc2, thisyields αCO = 4.6 M⊙ (K km s

−1 pc2)−1 and XCO,20 =2.1. Because the relation is not strictly linear αCO willchange by ∼ 60% for an order of magnitude change inluminosity (Fig. 2). Therefore GMCs with lower lumi-nosities (and masses) will have somewhat larger mass-to-light ratios and conversion factors than more luminousGMCs.

101

102

103

104

105

106

107

100

101

αCO[M

⊙(K

km

s−1pc2]−

1 )

LCO (K km s−1pc 2)

Fig. 2.— Relation between virial αCO and CO luminosity forGMCs in the Milky Way (Solomon et al. 1987). We have correctedthe numbers in the original table to reflect the updated distance tothe Galactic Center of 8.5 kpc. The dependence of αCO on LCOarises from the fact that the correlation between Mvir and LCO hasa nonlinear slope (Mvir ∝ LCO

0.815±0.013), following the expec-tations from Eq. 10 for approximately constant brightness tem-perature. This results in αCO ≈ 4.61 (LCO/10

5)−0.185, denotedby the red thick dashed line (the dispersion around this relationis ±0.15 dex). The nominal value at LCO = 10

5 K km s−1 pc2 isillustrated by the thin black dashed line.

Independent analysis using the same survey byScoville et al. (1987) yields a very similar mass-luminosity relation. After accounting for the differ-ent coefficients used for the calculation of the virialmass, the relation is Mvir = 33.5LCO

0.85. For aLCO ≈ 105 K km s−1 pc2 cloud this yields αCO =6.0 M⊙ (K km s

−1 pc2)−1 and XCO,20 = 2.8 (this workuses R⊙ = 8.5 kpc). Interestingly, there is no substantialdifference in the mass-luminosity relation for GMCs withor without HII regions (Scoville & Good 1989), althoughthe latter tend to be smaller and lower mass, and haveon average half of the velocity-integrated CO brightnessof their strongly star-forming counterparts. The result-ing difference in TB could have led to a displacement inthe relation, according to the simple reasoning leading toEq. 10, but it appears not to be significant.

4.1.3. Considerations and Limitations

Besides the already discussed applicability of the virialtheorem, there are a number of limitations to virial stud-ies. Some are practical, while others are fundamental tothe virial technique. On the practical side, virial studiesare sensitive to cloud definitions and biases induced bysignal-to-noise. These will impact both the values of Rand σ used to compute the mass. In noise-free measure-ments isolated cloud boundaries would be defined usingcontours of zero emission, when in reality it is neces-sary to define them using a higher contour (for example,Solomon et al. 1987, use a TB ∼ 4 K CO brightnesscontour). Scoville et al. (1987) discuss the impact of thiscorrection, studying the “curve of growth” for R and σas the definition contour is changed in high signal-to-

11

noise observations. Moreover, isolated clouds are rareand it is commonly necessary to disentangle many par-tially blended features along the line of sight. To measurea size clouds need to be resolved, and if appropriate thetelescope beam size needs to be deconvolved to estab-lish the intrinsic cloud size. This is a major concern inextragalactic studies, but even Galactic datasets are fre-quently undersampled which affects the reliability of theR and LCO determinations. Given these considerations,it is encouraging that two comprehensive studies usingindependent analysis of the same survey come to valuesof αCO that differ by only ∼ 30% for clouds of the sameluminosity.A fundamental limitation of the virial technique is that

CO needs to accurately sample the full potential and sizeof the cloud. For example, if because of photodissocia-tion or other chemistry CO is either weak or absent fromcertain regions, its velocity dispersion may not accuratelyreflect the mass of the cloud. This is a particular con-cern for virial measurements in low metallicity regions(see §6), although most likely it is not a limitation inthe aforementioned determinations of XCO in the innerGalaxy.

4.2. Column Density Determinations Using Dust andOptically Thin Lines

Perhaps the most direct approach to determining theH2 column density is to employ an optically thin tracer.This tracer can be a transition of a rare CO isotopologueor other chemical species (e.g., CH Magnani et al. 2003).It can also be dust, usually optically thin in emission atfar-infrared wavelengths, and used in absorption throughstellar extinction studies.

4.2.1. CO Isotopologues

A commonly used isotopologue is 13CO. Its abun-dance relative to 12CO is down by a factor approach-ing the 12C/13C≈ 69 isotopic ratio at the solar circle(12C/13C≈ 50 at RGC ≈ 4 kpc, the galactocentric radiusof the Molecular Ring) as long as chemical fractionationand selective photodissociation effects can be neglected(Wilson 1999). Given this abundance ratio and under theconditions in a dark molecular cloud 13CO emission maynot always be optically thin, as τ1 ∼ 1 requires AV ∼ 5.The procedure consists of inverting the observed inten-

sity of the optically thin tracer to obtain its column (orsurface) density. In the case of isotopologues, this columndensity is converted to the density of CO using the (ap-proximate) isotopic ratio. Inverting the observed inten-sity requires knowing the density and temperature struc-ture along the line of sight, which is a difficult problem. Ifmany rotational transitions of the same isotopologue areobserved, it is possible to model the line of sight columndensity using a number of density and temperature com-ponents. In practice an approximation commonly usedis local thermodynamic equilibrium (LTE), the assump-tion that a single excitation temperature describes thepopulation distribution among the possible levels alongthe line of sight. It is also frequently assumed that 12COand 13CO share the same Tex, which is particularly jus-tifiable if collisions dominate the excitation (Tex = Tkin,the kinetic temperature of the gas). Commonly used ex-pressions for determining N(13CO) under these assump-tions can be found in, for example, Pineda et al. (2010).

Note, however, that if radiative trapping plays an im-portant role in the excitation of 12CO, Tex for

13CO willgenerally be lower due to its reduced optical depth (e.g.,Scoville & Sanders 1987b).Dickman (1978) characterized the CO column density

in over 100 lines of sight toward 38 dark clouds, focus-ing on regions where the LTE assumption is unlikelyto introduce large errors. The combination of LTE col-umn densities with estimates of AV performed using starcounts yields AV ≈ (4.0±2.0)×10−16N(13CO) cm2 mag.Comparable results were obtained in detailed studiesof Taurus by Frerking, Langer & Wilson (1982, notethe nonlinearity in their expression) and Perseus byPineda, Caselli & Goodman (2008), the latter using a so-phisticated extinction determination (Lombardi & Alves2001). Extinction can be converted into molecular col-umn density, through the assumption of an effectivegas-to-dust ratio. Bohlin, Savage & Drake (1978) deter-mined a relation between column density and reddening(selective extinction) such that [N(HI)+2N(H2)]/E(B−V ) ≈ 5.8 × 1021 atoms cm−2 mag−1 in a survey of in-terstellar Lyα absorption carried out using the Coperni-cus satellite toward 75 lines of sight, mostly dominatedby HI. For a “standard” Galactic interstellar extinctioncurve with RV ≡ AV /E(B − V ) = 3.1, this results in

NH ≡ N(HI) + 2N(H2) ≈ 1.9× 1021 cm−2 AV . (21)A much more recent study using Far Ultraviolet Spectro-scopic Explorer observations finds essentially the samerelation (Rachford et al. 2009). In high surface densitymolecular gas RV may be closer to 5.5 (Chapman et al.2009), and Eq. 21 may yield a 40% overestimate(Evans et al. 2009). Using Eq. 21, the approxi-mate relation between 13CO J = 1 → 0 and molecu-lar column density is N(H2) ≈ 3.8 × 105 N(13CO).Pineda, Caselli & Goodman (2008) find a similar resultin a detailed study of Perseus, with an increased scatterfor AV & 5. Goldsmith et al. (2008) use these resultstogether with an averaging method to increase the dy-namic range of their 13CO and 12CO data, a physicallymotivated variable 12CO/13CO ratio, and a large veloc-ity gradient excitation analysis, to determine H2 columndensities in Taurus. They find that XCO,20 ≈ 1.8 recov-ers the molecular mass over the entire region mapped,while there is a marked increase in the region of lowcolumn density, where XCO increases by a factor of5 where N(H2) < 10

21 cm−2. As a cautionary noteabout the blind use of 13CO LTE estimates, however,Heiderman et al. (2010) find that this relation betweenH2 and

13CO underestimates N(H2) by factors of 4 − 5compared with extinction-based results in the Perseusand Ophiuchus molecular clouds.

4.2.2. Extinction Mapping

Extinction mapping by itself can be directly employedto determine XCO. It fundamentally relies on the as-sumption of spatially uniform extinction properties forthe bands employed, and on the applicability of Eq. 21to convert extinction into column density.Frerking, Langer & Wilson (1982) determined

XCO,20 ≈ 1.8 in the range 4 . AV . 12 in ρ Oph, whilethe same authors found constant W(CO) for AV & 2

12

in Taurus. Lombardi, Alves & Lada (2006) studied thePipe Nebula and found a best fit XCO in the rangeXCO,20 ≈ 2.9 − 4.2, but only for K-band extinctionsAK > 0.2 (equivalent to AV > 1.8, Rieke & Lebofsky1985). A simple fit to the data ignoring this nonlinearityyields XCO,20 ∼ 2.5. The Pineda, Caselli & Goodman(2008) study of Perseus finds XCO,20 ≈ 0.9 − 3 over anumber of regions. The relation between CO and H2,however, is most linear for AV . 4, becoming saturatedat larger line-of-sight extinctions.

Fig. 3.— Relation between CO column density and extinctionin the Taurus molecular cloud (Pineda et al. 2010). The figureshows the pixel-by-pixel relation between gas-phase CO columndensity (obtained from 13CO) and AV . The blue line illustratesthe “average” linear relation for 3 . AV . 10, N(

12CO) ≈1.01 × 1017AV cm

−2 (implying CO/H2 ≈ 1.1 × 10−4 for the as-sumed isotopic ratio). The linearity is clearly broken for AV & 10.(Pineda et al. 2010) show that linearity is restored to high AV af-ter applying a correction for CO freeze-out into dust grain mantles.

Pineda et al. (2010) extend the aforementionedGoldsmith et al. (2008) study of Taurus by character-izing the relation between reddening (from the TwoMicron All Sky Survey, 2MASS) and CO column density(derived from 13CO) to measure XCO,20 ≈ 2.1. Theyfind that the relation between AV and CO flattensfor AV & 10 (Fig. 3), a fact that they attribute tofreeze-out of CO onto dust grains causing the formationof CO and CO2 ice mantles. Including a correction forthis effect results in a linear relation to AV . 23. ForAV . 3 the column density of CO falls below the linearrelationship, likely due to the effects of photodissoci-ation and chemical fractionation. Along similar lines,Heiderman et al. (2010) find that in Ophiuchus andPerseus CO can underpredict H2 with respect to AV forΣmol > 200 M⊙ pc

−2 by as much as ∼ 30%.Paradis et al. (2012) recently used a high-latitude ex-

tinction map derived from 2MASS data using an exten-sion of the NICER methodology (Dobashi et al. 2008,2009) to derive XCO in sample of nearby clouds with|b| > 10◦. They find XCO,20 ≈ 1.67± 0.08 with a some-what higher value XCO,20 ≈ 2.28 ± 0.11 for the innerGalaxy region where |l| < 70◦. They report an excess inextinction over the linear correlation between total gas

and AV at 0.2 . AV . 1.5, suggestive of a gas phasethat is not well traced by either 21 cm or CO emission.We will return to this in §4.2.4.

4.2.3. Dust Emission

The use of extinction mapping to study N(H2) ismostly limited to nearby Galactic clouds, since it needs abackground stellar distribution, minimal foreground con-fusion, and the ability to resolve individual stars to deter-mine their reddening. Most interestingly, the far-infraredemission from dust can also be employed to map the gasdistribution. Indeed, dust is an extraordinarily egalitar-ian acceptor of UV and optical photons, indiscriminatelyprocessing them and reemitting in the far-infrared. Inprinciple, the dust spectral energy distribution can bemodeled to obtain its optical depth, τd(λ), which shouldbe proportional to the total gas column density underthe assumption of approximately constant dust emissiv-ity per gas nucleon, fundamentally the product of thegas-to-dust ratio and dust optical properties.How valid is this assumption? An analysis of the corre-

lation between τd and HI was carried out at high Galacticlatitudes by Boulanger et al. (1996), who found a typicaldust emissivity per H nucleon of

δDGR ≡ τd/NH ≈ 1.0× 10−25(λ/250µm)−β cm2, (22)with β = 2, in excellent accord with the recent valuefor high latitude gas derived using Planck observa-tions (Planck Collaboration et al. 2011c, who prefer β =1.8). They also identified a break in the correlation forN(HI) & 5× 1020 cm−2 suggestive of an increasingly im-portant contribution from H2 to NH , in agreement withresults from Copernicus (Savage et al. 1977). There isevidence that the coefficient in Eq. 22 changes in molec-ular gas. It may increase by factors of 2− 3 at very highcolumn densities (Schnee et al. 2008; Flagey et al. 2009;Planck Collaboration et al. 2011d), likely due to graingrowth or perhaps solid state effects at low temperatures(e.g., Mény et al. 2007). Note, however, that recent workusing Planck in the Galactic plane finds δDGR ≈ (0.92±0.05) × 10−25 cm2 at 250 µm, with no significant vari-ation with Galactic radius (Planck Collaboration et al.2011b). This δDGR is almost identical to that observedin dust mixed with mostly atomic gas at high latitudes,suggesting that the aforementioned emissivity variationsare very localized.This excellent correlation between τd and NH is the

basis for a number of studies that use dust emis-sion to determine H2 column densities. Most no-tably, Dame, Hartmann & Thaddeus (2001) employedthe Columbia survey of molecular gas in the Galacticplane together with the Dwingeloo-Leiden HI survey andthe IRAS temperature-corrected 100 µm spectral densitymap by Schlegel, Finkbeiner & Davis (1998). With thesedata, N(H2) can be obtained using

N(H2) = (τd/δDGR −N(HI))/2, (23)which simply states that the dust optical depth (τd) isa perfect tracer of the total column density of gas whenthe emissivity per nucleon (δDGR) is known. As we justdiscussed, δDGR can be straightforwardly determined onlines of sight dominated by atomic gas, for example. The

13

Fig. 4.— Planck results in the Aquila-Ophiuchus flare(Planck Collaboration et al. 2011a). The figure shows moleculargas column density with a color range N(H2) ≈ −1.5 × 1021

cm−2(dark blue) to 3.5 × 1021 cm−2(red). The black contoursillustrate the CO emission from the Columbia survey, with valuesICO ≈ 2, 10, 20 K km s

−1. The black region lacks CO information.

comparison of the molecular column density so derivedwith the observedW(CO) yieldsXCO,20 ≈ 1.8±0.3, validfor |b| > 5◦ on large scales across the Galaxy. This studyalso finds evidence for a systematic increase in XCO byfactors ∼ 2 − 3 at high Galactic latitude (b > 20◦), inregions with typical N(H2) . 0.5× 1020 cm−2 on ∼ 0.5◦angular scales.The excellent Planck dataset has afforded a new view

on this topic (Fig. 4). Planck Collaboration et al.(2011a) produced a new τd map for the Milky Way anda new determination of XCO using Eq. 23 for lines ofsight with |b| > 10◦. They obtain XCO,20 = 2.54± 0.13,somewhat larger than the previous study. This differenceis likely methodological, for example the use of differentfar-infrared wavelengths as well as local versus global cal-ibrations of δDGR.

4.2.4. CO-Faint Molecular Gas and Diffuse Lines of Sight

Most interestingly, Planck Collaboration et al. (2011a)observe a tight linear correlation between τd and N(HI)+2XCOW(CO) for AV . 0.4 and 2.5 . AV . 10, withan excess in τd in the intermediate range (Fig. 5).This excess can be understood in terms of a compo-nent of H2 (or possibly a combination of H2 and cold,opaque HI) that emits weakly in CO and is prevalent at0.4 . AV . 2.5 (the explanation is not unique, sincethe methodology cannot distinguish it from a changein dust emissivity in that narrow AV regime, but suchpossibility appears unlikely). This molecular componentarises from the region in cloud surfaces where gas is pre-dominantly H2 but most carbon is not in CO moleculesbecause the extinction is too low, essentially the PDRsurface (see §6.1 and Fig. 1). This component is fre-quently referred to as “CO-dark molecular gas” or some-times simply “dark gas” (Grenier, Casandjian & Terrier2005; Wolfire, Hollenbach & McKee 2010). In this re-view we will refer to it as “CO-faint,” which is a moreaccurately descriptive name. The existence of moleculargas with low CO abundance has been noted previouslyin theoretical models (e.g., van Dishoeck & Black 1988),in observations of diffuse gas and high-latitude clouds

(e.g., Lada & Blitz 1988), and in observations of irregu-lar galaxies (e.g., Madden et al. 1997). In this context,the results by Planck Collaboration et al. (2011a) are inqualitative agreement with already discussed observa-tions that show an increase in XCO at low molecular col-umn densities (e.g., Goldsmith et al. 2008; Paradis et al.2012).

Fig. 5.— Correlation between τd at 350 µm and total hy-drogen column density NH for XCO,20 = 2.3 (for |b| > 10

◦,Planck Collaboration et al. 2011a). The color scale represents thelogarithm of the number of lines-of-sight, and the blue dots the re-sult of NH binning. The red dashed lines indicate AV = 0.37 andAV = 2.5. The red solid line represents the best linear fit for lowNH. Note that it is also a good fit to the AV & 2.5 points. The ex-cess in the binned correlation over the red line for 0.37 . AV . 2.5is either an indication of “CO-faint” molecular gas, or possibly acombination of high optical-depth “opaque” HI with “CO-faint”H2, or a change in the dust emissivity over that AV regime.

The Planck observations are also in qualitative agree-ment with the analysis by Grenier, Casandjian & Terrier(2005), who correlated the diffuse γ-ray emission overthe entire sky with templates derived from the HI, CO,and dust, finding a component of gas not traced byCO evident in local clouds at high latitudes. Theiranalysis finds that this component is as important, bymass, as the “CO-bright” H2 component in several ofthese clouds, and increasingly more important for smallercloud masses. The recent analyses based on Fermi databy Abdo et al. (2010d) and Ackermann et al. (2012) arealso qualitatively compatible with these results, findingthat the “CO-faint” component amounts to 40%− 400%of the “CO-bright” mass in the Cepheus, Polaris, Chama-leon, R Cr A, and Cassiopeia clouds (small local molec-ular clouds).The ionized carbon far-infrared fine structure emission

provides an additional probe of molecular gas at low AV .Large scale [CII] observations of the (2P3/2 →2P1/2) fine-structure transition in the Milky Way and external galax-ies suggest it is due to a combination of emission from theCold Neutral Medium (CNM) and from PDRs located inthe surfaces of GMCs (Stacey et al. 1991; Shibai et al.1991; Bennett et al. 1994). The contribution from [CII]in the diffuse ionized gas, however, could also be im-portant (Heiles 1994; Madden et al. 1993), particularlyalong certain lines-of-sight (Velusamy et al. 2012).In regions where most of the emission arises in PDRs

[CII] has the potential to trace the “CO-faint” molec-

14

ular regime at low AV . Langer et al. (2010) analyze16 lines-of-sight in the plane of the Galaxy and findthat in about half of them the observed [CII] inten-sity can be entirely explained as due to carbon inatomic gas in the CNM. The other half, however, ex-hibits [CII]/N(HI) ratios that are too large to be dueto atomic gas and may have molecular to atomic ra-tios as large as N(H2)/N(HI) ∼ 6. While the verybrightest [CII] components investigated arise from dense(n > 105 cm−3) PDRs exposed to intense radiation fields,most of the [CII] emission in these molecular lines-of-sight can be explained as originating in the surfaces ofmodestly dense GMCs (n ∼ [3 − 300] × 103 cm−3) ex-posed to at most a few times the local interstellar ra-diation field at the solar circle (Pineda et al. 2010b).Following the reasoning in Velusamy et al. (2010) andLanger et al. (2010), a very approximate relation be-tween [CII] emission and “CO-faint” H2 is N(H2) ∼1.46× 1020W([CII]) − 0.35N(HI) cm−2, for W([CII]) inK km s−1 (see Eq. 7 to convert between Jy and K).We caution that the coefficients correspond to the MilkyWay carbon abundance and are very dependent on theassumed physical conditions, particularly the densities(we use n(HI) ∼ 200 cm−3, n(H2) ∼ 300 cm−3, andTHI ∼ TH2 ∼ 100 K).Liszt, Pety & Lucas (2010) measure XCO in diffuse

gas by first estimating the total hydrogen columnfrom dust continuum emission (using the map bySchlegel, Finkbeiner & Davis 1998), and subtracting theobserved HI column density (c.f., Eq. 23). They se-lect lines-of-sight with HCO+ absorption spectra againstbright extragalactic continuum sources. The CO emis-sion, together with the measured H2 column providesa measure of XCO. Surprisingly, Liszt, Pety & Lucas(2010) and Liszt & Pety (2012) find mean values in dif-fuse gas similar to those in GMCs. There are large vari-ations, however, about the mean with low XCO (brightCO) produced in warm T ∼ 100 K diffuse gas and highXCO (faint CO) produced at low N(H2) column densi-ties. The authors argue that these variations mainly re-flect the CO chemistry and its dependence on ultravioletradiation field, density, and total column density, ratherthan the H2 column density. We note that the observedCO column densities cannot be produced in steady-state PDR models (.e.g., Sonnentrucker et al. 2007). En-hanced CO production might occur through the “CH+

channel” driven by non-thermal ion-neutral reactions(Federman et al. 1996; Visser et al. 2009) or by pocketsof warm gas and ion-neutral reactions in turbulent dissi-pation regions (Godard, Falgarone & Pineau des Forets2009). Density fluctuations in a turbulent median mightalso increase the CO production (Levrier et al. 2012).Thus, although the mean XCO diffuse cloud value is simi-lar to GMCs, the CO emission from diffuse gas cannot beeasily interpreted as a measure of the molecular columnexcept perhaps in a statistical sense.

4.3. XCO Based on Gamma-Ray Observations

Diffuse γ-ray emission in the Galaxy is chiefly due tothree processes: neutral pion production and subsequentdecay in collisions between cosmic-rays and interstellarmatter, bremsstrahlung emission due to scattering ofcosmic-ray electrons by interstellar matter, and inverse

Compton scattering of low energy photons by cosmic-ray electrons. The first of these processes is the domi-nant production channel for diffuse γ-rays with energiesabove 200 MeV, although at high Galactic latitude therewill be an increasingly important inverse Compton com-ponent (Bloemen 1989). Interestingly, the fact that in-teractions between cosmic rays and nucleons give rise todiffuse γ-ray emission can be used to count nucleons inthe ISM, and indeed the use of XCO to represent the ra-tio N(H2)/W(CO) was introduced for the first time inγ-ray work using observations from the COS B satellite(Lebrun et al. 1983).Accounting for the pion decay and bremsstrahlung pro-

cesses, and neglecting the contribution from ionized gas,the basic idea behind modeling the emission is to usea relation Iγ =

∑

ǫγ,HI(Ri) [N(HI)i + 2 YCO W(CO)i](Bloemen 1989). Here Iγ is the diffuse γ-ray emis-sion along a line-of-sight, ǫγ is the HI gas emissiv-ity (a function of Galactocentric radius R), and YCOis a parameter that takes into account that emis-sivity in molecular clouds may be different than inatomic gas, due to cosmic-ray exclusion or concentration,XCO = YCO ǫγ,HI/ǫγ,H2 (Gabici, Aharonian & Blasi2007; Padovani, Galli & Glassgold 2009). This situationis analogous to that presented in dust emission tech-niques, where emissivity changes in the molecular andatomic components will be subsumed in the resultingvalue of XCO.Recent analyses use the approach Iγ ≈ qHIN(HI) +

qCOW(CO) + qEBVE(B − V )res + . . . , where the firstthree terms account for emission that is proportionalto the HI, the “CO-bright” H2, and a “CO-faint” H2component that is traced by dust reddening or emis-sion residuals (the additional terms not included ac-count for an isotropic γ-ray background and the con-tribution from point sources, e.g., Abdo et al. 2010d).The dust residual template consists of a dust map (e.g.,Schlegel, Finkbeiner & Davis 1998) with a linear combi-nation of N(HI) and W(CO) fitted and removed.

4.3.1. Observational Results

A discussion of the results of older analyses can befound in Bloemen (1989), here we will refer to a few ofthe more recent results using the Compton Gamma RayObservatory and Fermi satellites.Strong & Mattox (1996) analyzed the EGRET all-sky

survey obtaining XCO,20 ≈ 1.9±0.2. Similar results wereobtained by Hunter et al. (1997) for the inner Galaxy,and by Grenier, Casandjian & Terrier (2005) for cloudsin the solar neighborhood (see also §4.2.4). Strong et al.(2004) introduce in the analysis a Galactic gradient inXCO, in an attempt to explain the discrepancy betweenthe derived ǫγ(R) and the distributions of supernova rem-nants and pulsars, which trace the likely source of cos-mic rays in supernovae shocks. Matching the emissivityto the pulsar distribution requires a significant change inXCO between the inner and outer Galaxy (XCO,20 ≈ 0.4for R ∼ 2.5 kpc to XCO,20 ≈ 10 for R > 10 kpc).The sensitivity of Fermi has been a boon for stud-

ies of diffuse γ-ray emission in our Galaxy. Abdo et al.(2010d) and Ackermann et al. (2011, 2012b,d) analyzethe emission in the solar neighboorhood and the outerGalaxy. Taken together, they find a similar XCO forthe Local arm and the interarm region extending out to

15

R ∼ 12.5 kpc, XCO,20 ∼ 1.6 − 2.1 depending on the as-sumed HI spin temperature. The very local high-latitudeclouds in the Gould Belt have a lower XCO,20 ≈ 0.9, andappear not to represent the average properties at the so-lar circle (Ackermann et al. 2012b,c).It is important to note, however, that the Fermi

studies follow the convention XCO ≡ qCO/2qHI withthe parameters defined in the previous section. Thus,their definition of XCO does not include the “CO-faint” envelope that is traced by the dust residual tem-plate: it only accounts for the H2 that is emittingin CO. This is different from the convention adoptedin the dust studies discussed in the previous section,where all the H2 along a line-of-sight is associated withthe corresponding CO. Ackermann et al. (2012b) reportXCO,20 ≈ 0.96, 0.99, 0.63 for Chameleon, R CrA, andthe Cepheus/Polaris Flare region, respectively using theFermi convention. The ratio of masses associated withthe CO and the dust residual template (reported in theirTable 4) suggests that we should correct these numbersby factors of approximately 5, 2, and 1.4 respectively tocompare them with the dust modeling. This correctionresults in XCO,20 ≈ 4.8, 2, 0.9, bringing these clouds inmuch better (albeit not complete) agreement with theresults of dust modeling and the expectation that theybe underluminous — not overluminous — in CO due tothe presence of a large “CO-faint” molecular component.

4.3.2. Considerations and Limitations

We have already mentioned a limitation of γ-ray stud-ies of XCO, the degree to which they may be affected bythe rejection (or generation) of cosmic rays in molecularclouds. A major limitation of γ-ray determinations isalso the poor angular resolution of the observations. An-other source of uncertainty has become increasingly ap-parent with Fermi, which resolves the Magellanic Cloudsin γ-ray emission (Abdo et al. 2010,b). In the Clouds,the distribution of emission does not follow the distribu-tion of gas. Indeed, the emission is dominated by regionsthat are not peaks in the gas distribution, but may corre-spond to sites of cosmic ray injection (e.g., Murphy et al.2012), suggesting that better knowledge of the cosmic raysource distribution and diffusion will have an importantimpact on the results of γ-ray studies.Ackermann et al. (2012) carry out a thorough study

of the impact of systematics on the global γ-ray analy-ses that include a cosmic-ray generation and propagationmodel, frequently used to infer Galactic XCO gradients.They find that the XCO determination can be very sen-sitive to assumptions such as the cosmic ray source dis-tribution and the HI spin temperature, as well as theselection cuts in the templates (in particular, the dusttemplate). Indeed, the value in the outer Galaxy in theiranalysis is extremely sensitive to the model cosmic raysource distribution. The magnitude of a Galactic XCOgradient turns out to also be very sensitive to the un-derlying assumptions (see Fig. 25 in Ackermann et al.2012). What appears a robust result is a uniformly lowvalue of XCO near the Galactic center (R ∼ 0−1.5 kpc).This was already pointed out by much earlier γ-ray stud-ies (Blitz et al. 1985). The authors also conclude thatincluding the dust information leads to an improvementin the agreements between models and γ-ray data, evenon the Galactic plane, suggesting that “CO-faint” gas is

an ubiquitous phenomenon.

4.4. Synthesis: Value and Systematic Variations ofXCO in the Milky Way

There is an assuring degree of uniformity among thevalues of XCO obtained through the variety of method-ologies available in the Milky Way. Representative resultsfrom analyses using virial masses, CO isotopologues, dustextinction, dust emission, and diffuse γ-ray radiationhover around a “typical” value for the disk of the MilkyWay XCO ≈ 2×1020 cm−2(K km s−1)−1 (Table 1). Thisfact, combined with the simple theoretical argumentsoutlined in §2 as to the physics behind the H2-to-COconversion factor, as well as the results from elaboratenumerical simulations discussed in §3.2, strongly suggeststhat we know the mass-to-light calibration for GMCs inthe disk of the Milky Way to within ±0.3 dex certainly,and probably with an accuracy closer to ±0.1 dex (30%).This is an average number, valid over large scales. Indi-vidual GMCs will scatter around this value by a certainamount, and individual lines-of-sight will vary even more.There are, however, some systematic departures from

this value in particular regimes, as suggested by the sim-ple theoretical arguments. As pointed out by severalstudies, the Galactic center region appears to have anXCO value 3 − 10 times lower than the disk. In ad-dition to the aforementioned γ-ray results (Blitz et al.1985; Strong et al. 2004; Ackermann et al. 2012), a lowXCO in the center of the Milky Way was obtained byanalysis of the dust emission (Sodroski et al. 1995), andthe virial mass of its clouds (Oka et al. 1998, 2001). Wewill see in §5 that this is not uncommonly observed inthe centers of other galaxies. It is likely due to a com-bination of enhanced excitation (clouds are hotter) andthe dynamical effects discussed in §2.3.Departures may occur in some nearby, high-latitude

clouds, for example the Cepheus/Polaris Flare region. Acritical step to understand the magnitude or even ex-istence of such discrepancies will be to d